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NURBS solids by reparameterization methods
Gang Xu, Bernard Mourrain, André Galligo, Timon Rabczuk
To cite this version:
Gang Xu, Bernard Mourrain, André Galligo, Timon Rabczuk. High-quality construction of analysis-
suitable trivariate NURBS solids by reparameterization methods. Computational Mechanics, Springer
Verlag, 2014, 54 (5), pp.1303-1313. �10.1007/s00466-014-1060-y�. �hal-00922544v2�
(will be inserted by the editor)
High-quality construction of analysis-suitable trivariate NURBS solids by reparameterization methods
Gang Xu · Bernard Mourrain · Andr´ e Galligo · Timon Rabczuk
Received: date / Accepted: date
Abstract High-quality volumetric parameterization of computational domain plays an important role in three- dimensional isogeometric analysis. Reparameterization technique can improve the distribution of isoparametric curves/surfaces without changing the geometry. In this paper, using the reparameterization method, we inves- tigate the high-quality construction of analysis-suitable NURBS volumetric parameterization. Firstly, we intro- duce the concept of volumetric reparameterization, and propose an optimal M¨obius transformation to improve the quality of the isoparametric structure based on a new uniformity metric. Secondly, from given bound- ary NURBS surfaces, we present a two-stage scheme to construct the analysis-suitable volumetric parame- terization: in the first step, uniformity-improved repa- rameterization is performed on the boundary surfaces to achieve high-quality isoparametric structure with- out changing the shape; in the second step, from a new variational harmonic metric and the reparameterized boundary surfaces, we construct the optimal inner con- trol points and weights to achieve an analysis-suitable G. Xu
College of Computer Science, Hangzhou Dianzi University, Hangzhou 310018, P.R.China
E-mail: xugangzju@gmail.com B. Mourrain
Galaad, INRIA Sophia-Antipolis, 2004 Route des Lucioles, 06902 Cedex, France
E-mail: Bernard.Mourrain@inria.fr A. Galligo
University of Nice Sophia-Antipolis, 06108 Nice Cedex 02, France
E-mail: galligo@unice.fr T. Rabczuk
Institute of Structural Mechanics, Bauhaus-University Weimar, Marienstr. 15, D-99423 Weimar, Germany
E-mail: timon.rabczuk@uni-weimar.de
NURBS solid. Several examples with complicated ge- ometry are presented to illustrate the effectiveness of proposed methods.
Keywords Isogeometric analysis · Volumetric param- eterization · Boundary reparameterization · Uniformity metric
1 Introduction
The isogeometric analysis method proposed by Hughes et al. [14] employs the same type of spline represen- tation both for the geometry and for the physical so- lutions. This unified data representation allows for a seamless integration of the geometric design in numer- ical analysis. Moreover, the higher-order continuity of the (spline) basis function have advantageous over tra- ditional C
0finite element formulations based on La- grange polynomials which can be exploited in thin shell analysis [19] or weakly non-local continuum models.
The reduced number of parameters needed to describe the geometry is also of particular interest for shape and topology optimization.
Since isogeometric analysis was firstly proposed by
Hughes et al. [14] in 2005, many researchers working
on computational mechanical and geometric modeling
were involved in this field. We can classify the current
work on isogeometric analysis into four categories: (1)
isogeometric application in multi-physical problems [3,
9,13]; (2) application of different spline models in isoge-
ometric analysis [4,5,10,21,15]; (3) improving the accu-
racy and efficiency of IGA framework by refinement op-
erations and parallel computing [2,6,8,28,29]; (4) con-
structing analysis-suitable parameterization of compu-
tational domain from given boundary [1,18,20,26,29,
30,33,34].
The work presented this paper belongs to the fourth category. Mesh generation from a given CAD object is one of the most time-consuming step in a numerical analysis based on finite elements [14]. In the isogeo- metric analysis framework, the parameterization of the computational domain corresponds to the mesh gener- ation in finite element analysis and is the key of an effective isogeometric analysis that was developed for the main purpose of drastically shortening the model- ing time in numerical analysis.
Constructing analysis-suitable parameterization from a given CAD boundary representation remains one of the most significant challenges in isogeometric analysis.
In [6,28], the authors study the parametrization of com- putational domain in IGA, and show that the quality of parameterization has great impact on analysis results and efficiency. Pilgerstorfer and J¨ uttler show that in isogeometric analysis the condition number of the stiff- ness matrix, which is a key factor for the stability of the linear system, depends strongly on the quality of do- main parameterization [23]. Using volumetric harmonic functions, Martin et al. [18] proposed a fitting method for triangular mesh by B-spline parametric volumes.
Aigner et al. [1] proposed a variational approach to con- struct NURBS parameterization of swept volumes. In [12], a method is proposed to construct trivariate T- spline volumetric parameterization for genus-zero solid based on an adaptive tetrahedral meshing and mesh un- tangling technique. Zhang et al. proposed a robust and efficient approach to construct injective solid T-splines for genus-zero geometry from a boundary triangula- tion [33]. For mesh model with arbitrary topology, vol- umetric parameterization methods are proposed from the Morse theory [26] and Boolean operations [17].The input data of above methods is a triangle mesh. For the product modeling by CAD software, its boundary is usually in spline form. For parameterization prob- lem with spline boundary, Xu et al. proposed a con- straint optimization framework to construct analysis- suitable volume parameterization [30]. Spline volume faring is proposed by Pettersen and Skytt to obtain high-quality volume parameterization for isogeometric applications [24]. The construction of conformal solid T-spline from boundary T-spline representation is stud- ied by using octree structure and boundary offset [34].
In [31], variational harmonic method is proposed to con- struct analysis-suitable parameterization of computa- tional domain from given CAD boundary information.
Wang and Qian proposed an efficient method by com- bining divide-and-conquer, constraint aggregation and the hierarchical optimization technique to obtain valid trivariate B-spline solids from six boundary B-spline surfaces [27].
An analysis-suitable parameterization of computa- tional domain in isogeometric analysis should satisfy three requirements: 1) it should have no self-intersections, i.e, the mapping from the parametric domain to phys- ical domain should be injective; 2) the iso-parametric elements should be as uniform as possible; 3) the iso- parametric structure should be as orthogonal as pos- sible. Previous work mainly focus on the construction of inner control points [33,34,30,31,27]. To our best knowledge, the effect of the boundary parameteriza- tion on the interior volumetric parameterization has not been studied before. However, the quality of bound- ary parameterization has great effect on the subsequent volumetric parameterization results. Reparameteriza- tion technique can improve the quality of boundary pa- rameterization without changing the geometry. On the other hand, the weights in NURBS solid can be also considered as extra degree of freedom to obtain high- quality volumetric parameterization. In this paper, us- ing the reparameterization method, we investigate the high-quality construction of analysis-suitable NURBS volumetric parameterization. Our main contributions are:
– Two kinds of new volumetric metrics are introduced.
A uniformity metric is developed from the geomet- ric interpretation of the Jacobian and the concept of variance in statistics science; a new harmonic met- ric for volumetric smoothing is introduced from the variational formulation of harmonic equation.
– Volumetric spline reparameterization is introduced, and an optimal M¨ obius transformation is proposed to improve the uniformity of isoparametric struc- ture.
– By using uniformity-improved reparameterization of NURBS surfaces and the proposed variational har- monic metric, a two-stage scheme with multi-objective functions is proposed to construct the optimal in- ner control points and weights for analysis-suitable NURBS volumetric parameterization.
The rest of the paper is structured as follows. Af-
ter a new uniformity metric is introduced, Section 2
describes the optimal Mobius volumetric reparameter-
ization method for analysis-suitable NURBS solids. For
volumetric parameterization problem from given bound-
aries, a two-stage framework with multi-objective func-
tion is proposed to construct the optimal analysis-suitable
NURBS solid in Section 3. Some examples and com-
parisons are also presented in corresponding sections to
illustrate the effectiveness of the proposed methods. Fi-
nally, we conclude this paper and outline future works
in Section 4.
2 Volumetric reparameterization for analysis-suitable NURBS solids
2.1 Problem statement
The problem studied in this section can be stated as follows: given a trivariate NURBS solid, find an op- timal M¨ obius parameter transformation such that the isoparametric structure of reparameterized NURBS solid is as uniform as possible.
In the following subsection, we will review the def- inition of NURBS solids and introduce the Mobius vol- umetric transformation.
2.2 NURBS solids and Mobius volumetric transformation
A NURBS solid can be defined as follows,
S(u, v, w) = P
n i=0P
m j=0P
l k=0λ
i,j,kC
i,j,kN
ip(u)N
jq(v)N
kν(w) P
ni=0
P
m j=0P
l k=0λ
i,j,kN
ip(u)N
jq(v)N
kν(w) in which C
i,j,kare control points,λ
i,j,kare the weights, N
ip(u), N
jq(v) and N
kν(w) are B-spline basis function with degree p,q and ν respectively defined on the knot vectors
U = {0, · · · , 0, u
p+1, · · · , u
l, 1, · · · , 1}
V = {0, · · · , 0, v
q+1, · · · , v
m, 1, · · · , 1}
and
W = {0, · · · , 0, w
r+1, · · · , w
n, 1, · · · , 1}
Definition 1 (Mobius volumetric transformation) Sup- pose that α, β, γ ∈ [0, 1], the Mobius volumetric trans- formation can be defined as
u = (1 − α)ξ
α(1 − ξ) + (1 − α)ξ (1)
v = (1 − β)η
β(1 − η) + (1 − β)η (2)
w = (1 − γ)ζ
γ(1 − ζ) + (1 − γ)ζ (3)
After applying the Mobius transformation in (1)(2)(3) on the NURBS solid S(u, v, w), we can obtain a new
parametric representation S(ξ, η, ζ) of the NURBS solid e with the same control points as follows [16],
e S(ξ, η, ζ) = (x(ξ, η, ζ), y(ξ, η, ζ), z(ξ, η, ζ))
= P
n i=0P
m j=0P
l k=0e λ
i,j,kC
i,j,kN
ip(ξ)N
jq(η)N
kν(ζ) P
ni=0
P
m j=0P
l k=0e λ
i,j,kN
ip(ξ)N
jq(η)N
kν(ζ) in which the new weights
e λ
i,j,k= λ
i,j,kQ
p r=1K
i,rQ
q s=1L
j,sQ
ν t=1M
k,twith
K
i,r= (1 − α)(1 − u
i+r) + αu
i+r, (4)
L
j,s= (1 − β )(1 − v
j+s) + βv
j+s, (5) M
k,t= (1 − γ)(1 − w
k+t) + γw
k+t. (6) And the corresponding knot vectors are changed into U e = {0, · · · , 0,
| {z }
p+1
αu
p+1(1 − α)(1 − u
p+1) + αu
p+1, · · · , αu
l(1 − α)(1 − u
l) + αu
l, 1, · · · , 1
| {z }
p+1
}
V e = {0, · · · , 0,
| {z }
q+1
βv
q+1(1 − β )(1 − v
q+1) + βv
q+1, · · · , βv
m(1 − β)(1 − v
m) + βv
m, 1, · · · , 1
| {z }
q+1
}
and
W f = {0, · · · , 0,
| {z }
ν+1
γw
ν+1(1 − γ)(1 − w
ν+1) + γw
ν+1, · · · , γw
n(1 − γ)(1 − w
n) + γw
n, 1, · · · , 1
| {z }
ν+1
}.
2.3 Improving the uniformity of isoparametric structure by volumetric reparameterization
In this subsection, we will propose a uniformity-improved
volumetric reparameterization method based a new uni-
formity metric.
2.3.1 Uniformity metric for NURBS solids
In order to achieve a volumetric parameterization with uniform isoparametric structure, a new uniformity met- ric is firstly proposed. The uniform isoparametric struc- ture means that each isoparametric element has the same volume value. In probability theory and statistics, variance measures how far a set of numbers is spread out. A variance of zero indicates that all the values are identical. Hence, the uniformity of isoparametric struc- ture means that the variane between the volume value of each isoparametric element should be as small as pos- sible. Suppose that V
iis the volume of i-th element, and V
aveis the average element volume in the NURBS solid
e
S (ξ, η, ζ), the discrete variance σ
discan be defined as σ
dis= 1
N X
N i=0(V
i− V
ave)
2(7)
in which N is the number of sampling elements.
The Jacobian determinant can be considered as a scaling factor that relates the volume change of the parametric element to the physical element. Hence, the variance of element volume in Eq. (7) can be replaced by the variance of Jacobian determinant of the NURBS solid S e (ξ, η, ζ) , which can be defined in the form the continuous function as
σ = R
P
(det e J − J
mean)
2dP Z
P
dξdηdζ
(8)
in which J
meanis the average value of Jacobian deter- minant at each sampling point on the NURBS solid.
J
meancan be computed as the ratio between the vol- ume value of physical domain V
physicaland parametric domain V
parametric,
J
mean= V
physicalV
parametric= Z
P
e
S
ζ· ( S e
ξ× S e
η) dξdηdζ Z
P
dξdηdζ
= Z
P
det e J dξdηdζ Z
P
dξdηdζ
in which P is the parametric domain with knot vectors U, e V e and W. f e J is the Jacobian matrix of the NURBS solid S e (ξ, η, ζ) as follows,
J e =
x
ξx
ηx
ζy
ξy
ηy
ζz
ξz
ηz
ζ
(9)
Fig. 1
The uniformity metric illustrated by color-map.
As shown in [27,30], the uniformity is also related to the second order derivative of the parameterization. By combining the variance of Jacobian in (8), a new uni- formity metric at (ξ, η, ζ) can be defined as follows, µ( S e ) = (det e J−J
mean)
2+ω(k S e
ξξk
2+k S e
ηηk
2+k S e
ζζk
2),
(10) in which ω is a positive weight.
In order to show the effectiveness of the proposed metric, we present an example in Figure 1. The uni- formity metric is illustrated with color-map, which is rendered according to the value of µ( S e ). The red part has smallest value and the best uniformity, the blue part has the biggest value and the worst uniformity. We can find that the uniformity color-map is consistent with the size change of the isoparametric element.
2.3.2 Optimal volumetric Mobius reparameterization From the volumetric M¨ obius transformation and unifor- mity metric presented in previous sections, the optimal volumetric M¨ obius reparameterization problem can be stated as: given the initial parameterization S(u, v, w) of NURBS solid, find the optimal α, β, γ in the M¨obius volumetric transformation (4)(5)(6), such that the re- sulted parameterization S e (ξ, η, ζ) minimizes the follow- ing objective function
F
unif(α, β, γ) = Z
P
µ( S e ) dP (11)
in which µ(e S ) is defined in (10).
We solve this non-linear optimization problem with
the Levenberg-Marquardt method to obtain the values
of α, β and γ. The algorithm combines advantages of
the steepest descent method, in which minimization is
performed along the direction of the gradient, with the
Newton method, in which a quadratic model is used to
speed up the process of finding the minimum of a func-
tion. Hence, this algorithm obtained its operating sta-
bility from the steepest descent method, and adopted
its accelerated convergence in the minimum vicinity
from the Newton method.
(a) NURBS solid (b) Control lattice
(c) Initial isoparametric surfaces in
udirection
(d) Final isoparametric surfaces in
ξdirection
(e) Initial isoparametric surfaces in
vdirection
(f) Final isoparametric surfaces in
ηdirection
(g) Initial color-map of uniform metric
(h) Final color-map of uniform metric
Fig. 2
Volumetric M¨ obius reparameterization method for NURBS solid.
2.4 Example and comparison
Figure 2 shows an example and corresponding compar- ison result for volumetric M¨obius reparameterization.
The given NURBS solid and the control lattice are shown in Figure 2 (a) and 2 (b). Figure 2 (c) presents the initial isoparametric surfaces in u direction of the given NURBS volumetric parametrization. Figure 2 (d) shows the final isoparametric surfaces in ξ direction of the NURBS volumetric parametrization constructed by the M¨ obius reparameterization method with α = 0.323, β = 0.494 and γ = 0.217. The comparison of the isoparametric surfaces in v direction are also pre- sented in Figure 2 (e) and 2 (f). We use the uniformity colormap to show the uniformity of isoparametric struc- ture in the volume parameterizations. The uniformity colormap is computed according to the value of µ( S e ) defined in (10). From Fig.2(g) and Fig.2 (h) with the same scale, we can find that the volumetric parameteri- zation obtained by the optimal M¨ obius reparameteriza- tion method gives more uniform iso-parametric struc- ture than the initial given volume parameterization.
From the chain rule, we can directly prove that if initial volumetric parameterization has self-intersections, then M¨obius reparameterization method can not re- move the self-intersections. In the following section, we will propose a two-stage scheme to construct high-quality volumetric parameterization without self-intersections by boundary reparameterization.
3 Constructing analysis-suitable NURBS solids by boundary reparameterization
3.1 Main framework
Suppose that S is a simply connected bounded domain in three dimensional space with Cartesian coordinates (x; y; z)
T, and is bounded by six NURBS surfaces. The volume parameterization problem of three-dimensional computational domain in isogeometric analysis can be stated as: given six boundary NURBS surfaces, find the optimal inner control points and weights such that the resulting trivariate NURBS parametric volume is a good computational domain for 3D isogeometric anal- ysis.
The quality of boundary parameterization has great effect on the subsequent volumetric parameterization of computational domain. In this section, we will present a two-stage scheme to construct analysis-suitable NURBS solids: in the first step, boundary surface reparame- terization is performed to improve the quality of the boundary isoparametric structure; in the second step,
from the reparameterized boundary surfaces, we con- struct the optimal inner control points and weights to achieve an analysis-suitable NURBS solid.
3.2 boundary reparameterization
The boundary reparameterization in this part can be viewed as the degenerated case of the volumetric repa- rameterization in Section 2. For each given boundary NURBS surface
R(u, v) = P
n i=0P
m j=0λ
i,jC
i,jN
ip(u)N
jq(v) P
ni=0
P
m j=0λ
i,jN
ip(u)N
jq(v),
The following Mobius transformation is performed on R(u, v)
u = (1 − α)ξ α(1 − ξ) + (1 − α)ξ v = (1 − β)η
β(1 − η) + (1 − β)η
in which α, β ∈ [0, 1]. Then we can obtain a new para- metric NURBS surface R(ξ, η) with the same geometry e as R(u, v). R(ξ, η) has the same control points but dif- e ferent weights with R(u, v). The new weights e λ
i,j,kcan be computed from the old weights λ
i,j,kas follows, e λ
i,j,k= λ
i,j,kQ
p r=1K
i,rQ
q s=1L
j,swith K
i,rand L
j,sdefined in (4) and (5).
Then we seek for the optimal parameter α and β , such that the isoparametric net of resulting NURBS surface R(ξ, η) is as uniform as possible. That is, find e the optimal α and β to minimize the following objective function as shown in Section
Z
P
(det e J − J
avg)
2+ ω
1(k R e
ξξk
2+ k R e
ηηk
2) dξdη, (12) in which
J
avg= R
P
det J dξdη R
P
dξdη , and
e J =
x
ξx
ηy
ξy
η.
Similar with the trivariate case, the Levenberg-
Marquardt method is used to solve this nonlinear opti-
mization problem.
(a)
(b)
(c)
Fig. 3
Reparameterization example of a planar NURBS sur- faces: (a) NURBS surface and control mesh; (b)initial isopara- metric net on the surface; (c) final isoparametric net after optimal M´ obius reparameterization.
Figure 3 presents an example of boundary surface reparameterization. Figure 3 (a) presents the given pla- nar NURBS surface and its control mesh. Figure 3(b) presents the initial isoparametric net on the surface; the corresponding iso-parametric structure of the reparam- eterized NURBS surface obtained by optimal M¨ obius transformation is shown in Figure 3(c). Obviously, more uniform iso-parametric structure can be achieved with- out changing the boundary shape by optimal reparam- eterization technique.
3.3 Initial construction of NURBS solids
After boundary reparameterization, we need to con- struct the initial control points and weights for the subsequent optimization process. In [30], the discrete Coons method is proposed to construct B-spline volume by linear combination of boundary control points. In this approach, the compatible boundary surfaces with the same degree, knot vectors and the number of con- trol points are required. However, in practice, such re- quirements are usually not satisfied. Hence, some pre-
processing operation must be performed for the given opposite NURBS surfaces according to the following operation procedure:
– make the given opposite NURBS surfaces have the consistent parametric direction
– perform degree elevation to have the same degree – perform knot insertion to have the same number of
control points
When all the opposite surfaces on the boundary are compatible, the discrete coons method can be employed to construct the NURBS volumes [28][11] . That is, the interior control points C
i,j,kand weights λ
i,j,kcan be constructed as linear combination of boundary control points and weights. If we introduce the four-dimensional notation P
i,j,k= ( C
i,j,k, λ
i,j,k), the corresponding con- struction formula can be written as
Pi,j,k
= (1
−i/l)P0,j,k+
i/lPl,j,k+ (1
−j/m)Pi,0,k+j/m
Pi,m,k+ (1
−k/n)Pi,j,0+
k/nPi,j,n−
[1
−i/l, i/l]P0,0,kP0,m,k
Pl,0,k Pl,m,k
1
−j/m j/m
−
[1
−j/m, j/m]Pi,0,0 Pi,0,n Pi,m,0Pi,m,n
1
−k/n k/n
−
[1
−k/n, k/n]P0,j,0 Pl,j,0
P0,j,n Pl,j,n
1
−i/l i/l
+(1
−k/n)[1
−i/l, i/l]P0,0,0P0,m,0 Pl,0,0 Pl,m,0
1
−j/m j/m
+k/n
[1
−i/l, i/l]P0,0,nP0,m,n
Pl,0,n Pl,m,n
1
−j/m j/m
As shown in [30], the initial NURBS solid con- structed by discrete Coons method may have self-intersections and low quality. In the following we will propose a method to construct the optimal inner control points and weights to achieve an analysis-suitable NURBS solid based on a new variational harmonic metric.
3.4 Construction of analysis-suitable NURBS solids 3.4.1 Variational harmonic metric
The proposed volumetric parameterization method is based on the concept of harmonic mapping, which is a one-to-one transformation for three-dimensional do- mains. From the harmonic mapping theory, if f : S 7→
P is a harmonic mapping from S to P , then the inverse mapping f
−1: P 7→ S should be bijective.
The mapping f : S 7→ P is called harmonic map- ping, if f satisfies
∆ξ(x, y, z) = ξ
xx+ ξ
yy+ ξ
zz= 0 (13)
∆η(x, y, z) = η
xx+ η
yy+ η
zz= 0 (14)
∆ζ(x, y, z) = ζ
xx+ ζ
yy+ ζ
zz= 0 (15)
Different from the method in [31], in this paper, a new harmonic metric is proposed based on the varia- tional formulation of the PDEs (13)(14)(15), which is the classical Dirichlet integral as follows,
G = Z
Ω
(∇ξ)
2+ (∇η)
2+ (∇ζ)
2dxdydz (16) The above function can be transformed from physical domain to the parametric domain using Jacobian trans- formation, which is denoted as variational harmonic metric as follows,
F
harmonic= Z
P
[
13(k S
ξk
2+ k S
ηk
2+ k S
ηk
2)]
32det J dP
= Z
P
[
13tr(J
TJ)]
32det J dξdηdζ (17)
in which J is the Jacobian matrix of S (ξ, η, ζ) as defined in Eqn. (9).
3.4.2 Multi-objective optimization method for volumetric parameterization
The orthogonality of iso-parametric structure is also a key quality measure of analysis-suitable volumetric parameterization in numerical simulation [35]. The or- thogonality measure can be defined according to the differential geometry property of parametric volumes as follows ,
F
orth= Z
P
k S
ξ· S
ηk
2+ k S
η· S
ζk
2+ k S
ξ· S
ζk
2dP . (18) By combining the metrics defined in (11)(17)(18), a nonlinear optimization problem with multi-objective functions is achieved as follows
Ci,j,k
min
,λi,j,k(1 − θ
1− θ
2)F
harmonic+ θ
1F
unif+θ
2F
orth(19) where C
i,j,kand λ
i,j,kare control points and weights as design variables to be solved, θ
1and θ
2are weights for the balance between the harmonic metric, uniformity metric and orthogonality metric.
Since the problem in (19) is usually a large-scale optimization problem, we adapt L-BFGS method to obtain the optimal solution, which is a quasi-Newton method for solving unconstrained nonlinear minimiza- tion problems. In L-BFGS framework, the inverse Hes- sian matrix of the objective function is approximated by a sequence of gradient vectors from previous itera- tions. For more details, the reader can refer to [22].
(a) Boundary NURBS surfaces
(b) Boundary NURBS curves
(c) Initial boundary parameterization
(d) Optimized boundary parameterization
(e) Control lattice (f) Final isoparametric
structure
Fig. 4
Volumetric parameterization of human body model.
3.5 Experimental results
Figure 4 shows an example for volumetric parameter-
ization of human body. The given boundary NURBS
surfaces and curves are shown in Figure 4 (a) and Fig-
ure 4 (b). Figure 4 (c) presents initial isoparametric
net on one of the given boundary NURBS sufaces. Fig-
ure 4 (d) shows the uniformity-improved isoparametric
net on the reparameterized boundary surface by opti-
mal M¨ obius transformation. The control lattice of the
final NURBS volumetric parameterization is shown in
Figure 4 (e). To illustrate the quality of the parame-
terization, the iso-parametric surfaces of the resulting
NURBS volume are presented in Figure 4 (f). More vol-
(a) (b)
(c) (d)
Fig. 5
Volumetric parametrization of thumb model : (a) boundary NURBS surfaces and control mesh; (b) boundary NURBS curves; (c)resulting control lattice; (d) final isopara- metric structure.
Table 1
Quantitative data in Figure 2, Figure 4 , Figure 5 and Figure 6. # deg.: degree of B-spline parameterization; # con.: number of control points; # iter.: number of optimiza- tion iterations.
Example # Deg. # Con. #Iter.
Figure 2 4 8
×8
×8 4
Figure 4 3 13
×14
×13 13
Figure 5 3 8
×10
×13 9
Figure 6 3 8
×14
×11 11
umetric parameterization examples with complex ge- ometry are shown in Figure 5 and Figure 6.
Quantitative data of the examples presented in Fig- ure 2, Figure 4, Figure 5 and Figure 6 are summarized in Table 1. Overall, the volumetric parameterization obtained by the proposed two-stage method has high- quality, and is suitable for isogeometric applications.
4 Conclusion
The quality of boundary parameterization has great effect on the subsequent volumetric parameterization results. Reparameterization methods can improve the
quality of boundary parameterization without changing the geometry. In this paper, NURBS volumetric repa- rameterization is introduced into isogeometric analysis by using optimal M¨ obius transformation, and then the boundary surface reparameterization is performed as a pre-processing before constructing the inner control points and weights. Moreover, new uniformity metric and variational harmonic metric are also proposed for analysis-suitable volumetric parameterization. Experi- mental results illustrate that based on the reparame- terization methods, we can obtain high-quality NURBS volumetric parameterization results, which are suitable for subsequent isogeometric analysis.
In the future, we will study the piece-wise repa- rameterization method for high-quality NURBS volu- metric parameterization of computational domain [32].
The application of reparameterization technique in iso- geometric solving on NURBS surfaces is also a part of our future work.
Acknowledgment
The first author is partially supported by the National Nature Science Foundation of China (Nos. 61004117, 61272390, 61211130103), the Defense Industrial Tech- nology Development Program ( A3920110002), the Sci- entific Research Foundation for the Returned Overseas Chinese Scholars from State Education Ministry.
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(a) Boundary NURBS surfaces (b) Initial boundary parameterization (c) Optimized boundary parameterization
(d) Control lattice (e) Final isoparametric structure (top view)
Fig. 6
